Exercise 3.3 1.Solve the following pair of linear equations by the substitution method : Sol. (i) The given system of equations is x +3 = 14 …(1) and, x-y = 4 …(2) Substituting y = 14-x in (2), we get x – (14 -x) = 14 => x-14 x = 4=> 2x = 4 + …
Exercise 3.2 1.Form the pair of linear equations in the following problems, and find •their solution graphically. Hence, the cost of one pencil is Rs 3 and that of one pen is Rs 5.Verification: Put x = 3 and y = 5 in (1) and (2), we find that both the equations are satisfied. 2.On …
EXERCISE 3.1 1.Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be”. (Isn’t this interesting) ? Represent this situation algebraically and graphically. Sol.Let us denote the present age of daughter and her …
PAIR OF LINEAR EQUATIONSIN TWO VARIABLES 1.Two linear equations in the same two variables are called a pair of linear equations in two variables, or briefly, a linear pair. The most general form of a linear pair is a1x,+b1y +c1=0 a2x +b2y + C2 = 0 where a1, a2, b1, b2, C1, C2 are real …
EXERCISE 2.4 1.Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case: (i) 2×2+ x2 – 5x + 2; , 1,-2 (ii) x3 – 4×2 + 5x – 2; 2, 1, 1 Sol. (i) Comparing the given polynomial …
EXERCISE 2.3 1.Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following: (i) p(x) = x3-3×2+ 5x – 3, g (x) = x2 _ 2(ii) p(x) =x3 + 3×2+ 4x, g(x) = x2 + 1-x(iii) p(x) = x4– 5x+ 6, g(x) = 2 – x2 Sol. …
EXERCISE 2.2 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients: (i) x2 – 2x – 8 (ii) 4s2 – 4s + 1 (iii) 6×2 – 3 – 7x (iv)4u2 +8u (v) t2 – 15 (vi) 3×2 – x – 4 Sol. (i) We have, x2 – 2x – 8 = x2 – 2x …
EXERCISE 2.1 1. The graphs of y = p(x) are given in figures below for some polynomials p(x). Find the number of zeroes of p(x), in each case. Sol. (i) There are no zeroes as the graph does not intersect the x-axis. (ii) The number of zeroes is one as the graph intersects the x-axis …
POLYNOMIALS 1. An algebraic expression p(x) of the form p(x) = a0 , + a1x+a2 x2+ … + anxn, where a0, a1, a2, …, an are real numbers and all the indices of x are non-negative integers, is called a polynomial in x and the highest index n is called the degree of the polynomial, if an 0 …
EXERCISE 1.4 1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion : Sol. We know that if the denominator of a rational number has no prime factors other than 2 or 5, then it is expressible as a terminating, …